Why
is it that the number of petals in a flower is often one of the
following numbers: 3, 5, 8, 13, 21, 34 or 55? For example, the lily has
three petals, buttercups have five of them, the chicory has 21 of them,
the daisy has often 34 or 55 petals, etc. Furthermore, when one
observes the heads of sunflowers, one notices two series of curves, one
winding in one sense and one in another; the number of spirals not
being the same in each sense. Why is the number of spirals in general
either 21 and 34, either 34 and 55, either 55 and 89, or 89 and 144?
The same for pinecones : why do they have either 8 spirals from one
side and 13 from the other, or either 5 spirals from one side and 8
from the other? Finally, why is the number of diagonals of a pineapple
also 8 in one direction and 13 in the other?

Are
these numbers the product of chance? No! They all belong to the
Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. (where
each number is obtained from the sum of the two preceding). A more abstract way of putting
it is that the Fibonacci numbers fn are given by the formula
f1 = 1, f2 = 2, f3 = 3, f4 =
5 and generally f n+2 = fn+1 + fn . For a long
time, it had been noticed that these numbers were important in nature,
but only relatively recently that one understands why. It is a question
of efficiency during the growth process of plants (see below).

The
explanation is linked to another famous number, the golden mean, itself
intimately linked to the spiral form of certain types of shell. Let's
mention also that in the case of the sunflower, the pineapple and of
the pinecone, the correspondence with the Fibonacci numbers is very
exact, while in the case of the number of flower petals, it is only
verified on average (and in certain cases, the number is doubled since
the petals are arranged on two levels).

Let's
underline also that although Fibonacci historically introduced these
numbers in 1202 in attempting to model the growth of populations of
rabbits, this does not at all correspond to reality! On the contrary,
as we have just seen, his numbers play really a fundamental role in the
context of the growth of plants

THE
EFFECTIVENESS OF THE GOLDEN MEAN

The
explanation which follows is very succinct. For a much more detailed
explanation, with very interesting animations, see the web site in the
reference.

In
many cases, the head of a flower is made up of small seeds which are
produced at the centre, and then migrate towards the outside to fill
eventually all the space (as for the sunflower but on a much smaller
level). Each new seed appears at a certain angle in relation to the
preceeding one. For example, if the angle is 90 degrees, that is 1/4 of
a turn, the result after several generations is that represented by
figure 1.

Of course, this
is not the most efficient way of filling space. In fact, if the angle
between the appearance of each seed is a portion of a turn which
corresponds to a simple fraction, 1/3, 1/4, 3/4, 2/5, 3/7, etc (that is
a simple rational number), one always obtains a series of straight
lines. If one wants to avoid this rectilinear pattern, it is necessary
to choose a portion of the circle which is an irrational number (or a
nonsimple fraction). If this latter is well approximated by a simple
fraction, one obtains a series of curved lines (spiral arms) which even
then do not fill out the space perfectly (figure 2).

In
order to optimize the filling, it is necessary to choose the most
irrational number there is, that is to say, the one the least well
approximated by a fraction. This number is exactly the golden mean. The
corresponding angle, the golden angle, is 137.5 degrees. (It is
obtained by multiplying the non-whole part of the golden mean by 360
degrees and, since one obtains an angle greater than 180 degrees, by
taking its complement). With this angle, one obtains the optimal
filling, that is, the same spacing between all the seeds (figure 3).

This
angle has to be chosen very precisely: variations of 1/10 of a degree
destroy completely the optimization. (In fig 2, the angle is 137.6
degrees!) When the angle is exactly the golden mean, and only this one,
two families of spirals (one in each direction) are then visible: their
numbers correspond to the numerator and denominator of one of the
fractions which approximates the golden mean : 2/3, 3/5, 5/8, 8/13,
13/21, etc.

These
numbers are precisely those of the Fibonacci sequence (the bigger the
numbers, the better the approximation) and the choice of the fraction
depends on the time laps between the appearance of each of the seeds at
the center of the flower.

This
is why the number of spirals in the centers of sunflowers, and in the
centers of flowers in general, correspond to a Fibonacci number.
Moreover, generally the petals of flowers are formed at the extremity
of one of the families of spiral. This then is also why the number of
petals corresponds on average to a Fibonacci number.

REFERENCES:

1.
Ron Knot's excellent internet site at the University of Surrey on this and
related topics.